Spectrahedron

A spectrahedron

In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of n × n positive semidefinite matrices forms a convex cone in R^{n × n}, and a spectrahedron is a shape that can be formed by intersecting this cone with a linear affine subspace.

Spectrahedra are the solution spaces of semidefinite programs. Every spectrahedron is a convex set and also semialgebraic, but the reverse (conjectured to be true until 2017) is false.

An example of a spectrahedron is the spectraplex, defined as

$\mathrm {Spect} _{n}=\{X\in \mathbf {S} _{+}^{n}\mid \mathrm {Tr} (X)=1\}$

where $\mathbf {S} _{+}^{n}$is the set of n × n positive semidefinite matrices and $\mathrm {Tr} (X)$ is the trace of the matrix $X$. The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.